Incomplete Multiplicative Preference Relations Research Articles

Heterogeneous group decision making in the setting of incomplete preference relations

This paper focuses on heterogeneous group decision making (GDM) in the setting of incomplete preference relations, including incomplete fuzzy preference relations, incomplete multiplicative preference relations, incomplete additive linguistic preference relations, and incomplete multiplicative linguistic preference relations. Following the consistency-consensus optimization analysis, a new GDM method is proposed which can address inconsistent and incomplete preference relations. Firstly, we consider how to determine unknown information and how to obtain consistent preference relations. In contrast to the existing methods which rank the alternatives from fuzzy preference relations using the additive or multiplicative consistency analysis, this paper follows the consistency level to decide to use the additive or multiplicative consistency. Following the consensus analysis for each type of consistent preference relations, we calculate their comprehensively consistent preference relations and the priority vectors. After that, we build a consistency-based model for calculating the comprehensive priority vector. Furthermore, we offer an ordered position consensus-based method using the built optimization models. The proposed heterogeneous GDM method with incomplete preference relations provides us with a very useful way for dealing with GDM problems in heterogeneous environments.

Group decision making based on incomplete multiplicative and fuzzy preference relations

The main aim of this paper is to investigate the group decision making on incomplete multiplicative and fuzzy preference relations without the requirement of satisfying reciprocity property. This paper introduces a new characterization of the multiplicative consistency condition, based on which a method to estimate unknown preference values in an incomplete multiplicative preference relation is proposed. Apart from the multiplicative consistency property among three known preference values, the method proposed also takes the multiplicative consistency property among more than three values into account. In addition, two models for group decision making with incomplete multiplicative preference relations and incomplete fuzzy preference relations are presented, respectively. Some properties of the collective preference relation are further discussed. Numerical examples are provided to make a discussion and comparison with other similar methods.

An approach to incomplete multiplicative preference relations and its application in group decision making

To measure the consistency of multiplicative preference relations, a reasonable concept of consistent multiplicative preference relations is defined, and the multiplicative geometric consistent index (MGCI) is presented. In a simulation method, the average values of the MGCI are studied. Considering incomplete multiplicative preference relations, a consistency-based linear programming model is constructed, which is optimal from the point of view of consistency with respect to the known judgments. In a group decision making context, the group consensus index (GCOI) is given to measure the consensus of individual multiplicative preference relations. Then, the hybrid weighted geometric mean (HWGM) operator is defined to calculate the collective multiplicative preference relation. Based on the consistency and consensus analysis, an approach to group decision making with incomplete multiplicative preference relations is developed. Furthermore, we briefly research the application of the new method to incomplete interval multiplicative preference relations and incomplete fuzzy preference relations. Meanwhile, the associated examples are offered to show the practicability and efficiency of the proposed procedure.

A consensus reaching process under incomplete multiplicative preference relations

Multiplicative preference relation is the most common tool used to express decision makers' preferences over pairs of criteria. Due to the decision makers' own personal experience, learning, situation, state of mind, or the complexity of the considered decision-making problems, in many cases, the decision makers cannot or are unwilling to provide their preferences over some pairs of criteria, and thus may construct incomplete multiplicative preference relations in which some of the elements are missing, we call these problems group decision-making problems with incomplete multiplicative preference relations. In this paper, we first investigate the approach to extending each incomplete multiplicative preference relation to a complete multiplicative preference relation by making use of the multiplicative transitivity properties. Then, we present some deviation measures to evaluate the consensus among the individual multiplicative preference relations and their collective multiplicative preference relation. On the basis of the above results, finally, we develop a consensus reaching process of group decision making with incomplete multiplicative preference relations, which establishes a sound theoretical basis for the priorities of criteria.

On group decision making with four formats of incomplete preference relations

In this paper, a new approach is proposed to solve group decision making (GDM) problems where the preference information on alternatives provided by decision makers (DMs) is represented in four formats of incomplete preference relations, i.e., incomplete multiplicative preference relations, incomplete fuzzy preference relations, incomplete additive linguistic preference relations, incomplete multiplicative linguistic preference relations. In order to make the collective opinion close each decision maker’s opinion as near as possible, an optimization model is constructed to integrate the four different formats of incomplete preference relations and to compute the collective ranking values of the alternatives. The ranking of alternatives or selection of the most desirable alternative(s) is directly obtained from the derived collective ranking values. A numerical example is also used to illustrate the applicability of the proposed approach.

Nonlinear Programming Model Integrating Different Preference Structures

In this correspondence paper, we consider a large group decision-making problem in which the decision makers provide their preferences over a number of alternatives using distinct preference structures, including the following: 1) utility values; 2) preference orderings; 3) multiplicative preference relations; 4) incomplete multiplicative preference relations; 5) fuzzy preference relations; and 6) incomplete fuzzy preference relations. We establish a nonlinear programming model to integrate all of these preference structures and then employ a genetic algorithm to find the solution to the problem. This model can be reduced to a variety of special models suitable for group decision-making situations in which the preferences are represented in one (or several) of the aforementioned preference structures. Finally, detailed numerical analysis is provided to verify the feasibility and effectiveness of the model.

An integrated model-based interactive approach to FMAGDM with incomplete preference information

Incomplete fuzzy preference relations, incomplete multiplicative preference relations, and incomplete linguistic preference relations are very useful to express decision makers' incomplete preferences over attributes or alternatives in the process of decision making under fuzzy environments. The aim of this paper is to investigate fuzzy multiple attribute group decision making problems where the attribute values are represented in intuitionistic fuzzy numbers and the information on attribute weights is provided by decision makers by means of one or some of the different preference structures, including weak ranking, strict ranking, difference ranking, multiple ranking, interval numbers, incomplete fuzzy preference relations, incomplete multiplicative preference relations, and incomplete linguistic preference relations. We transform all individual intuitionistic fuzzy decision matrices into the interval decision matrices and construct their expected decision matrices, and then aggregate all these expected decision matrices into a collective one. We establish an integrated model by unifying the collective decision matrix and all the given different structures of incomplete weight preference information, and develop an integrated model-based approach to interacting with the decision makers so as to adjust all the inconsistent incomplete fuzzy preference relations, inconsistent incomplete linguistic preference relations and inconsistent incomplete multiplicative preference relations into the ones with acceptable consistency. The developed approach can derive the attribute weights and the ranking of the alternatives directly from the integrated model, and thus it has the following prominent characteristics: (1) it does not need to construct the complete fuzzy preference relations, complete linguistic preference relations and complete multiplicative preference relations from the incomplete fuzzy preference relations, incomplete linguistic preference relations and incomplete multiplicative preference relations, respectively; (2) it does not need to unify the different structures of incomplete preferences, and thus can simplify the calculation and avoid distorting the given preference information; and (3) it can sufficiently reflect and adjust the subjective desirability of decision makers in the process of interaction. A practical example is also provided to illustrate the developed approach.

A goal programming approach to group decision-making with three formats of incomplete preference relations

This paper proposes a goal programming approach to solve the group decision-making problem where the preference information about alternatives provided by decision makers can be represented in three formats, i.e., incomplete multiplicative preference relations, incomplete fuzzy preference relations and incomplete linguistic preference relations. In the approach, a transformation function is introduced to transform the incomplete linguistic preference relation into an incomplete fuzzy preference relation. To narrow the gap between the collective opinion and each decision maker’s opinion, a liner goal programming model is constructed to integrate the three different formats of incomplete preference relations and to compute the collective ranking values of the alternatives. Thus, the ranking order of alternatives or selection of the most desirable alternative(s) is obtained directly according to the computed collective ranking values. A numerical example is also used to illustrate the feasibility and the applicability of the proposed approach.